% Presentation by 郭睿明@四川大学
% Presentation by dandelight@SCU
% This presentation is really hard-core, please listen intently.
% Time limit: 15mins.
% For personal discussion only.
% Imformation on paper:
% Title:

\documentclass[12pt, aspectratio=169]{beamer}
\usetheme{PaloAlto}
\usecolortheme{crane}
\title{Single Image Optical Flow Estimation\\ with an Event Camera}
% \institute{Sichuan University}
\usefonttheme{serif}

\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}

\newcommand{\iee}{i.\,e.}

\newcommand{\vB}{\mathbf{B}}
\newcommand{\vK}{\mathbf{K}}
\newcommand{\vk}{\mathbf{k}}
\newcommand{\vL}{\mathbf{L}}
\newcommand{\vE}{\mathbf{E}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vX}{\Omega}%\mathbf{X}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vI}{\mathbf{I}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vq}{\mathbf{q}}
\newcommand{\prox}{\mathcal{P}}
\newcommand{\rmT}{\mathrm{T}}
\newcommand{\abs}{\mathrm{abs}}
\newcommand{\divs}{\mathrm{div}}
\newcommand{\R}{{\rm I} \! {\rm R}}


\begin{document}

  \AtBeginSection[] {
    \begin{frame}<beamer>
      \tableofcontents[currentsection]
    \end{frame}
  }

  \begin{frame}
    \titlepage
  \end{frame}

  \section{Introduction}
  \begin{frame}{Introduction}{Event-based Cameras}
    Frame based cameras are bio-inspired sensors that respond to per-pixel intensity changes.

    \vspace{5mm}

    Unlike conventional cameras, which record images at a fixed rate, event cameras trigger events whenever the change in intensity at pixel level exceeds a preset threshold, namely, $c$.

    \begin{align} %\label{eq:log}
      \left|\log ( {\vL(\vx,t)})-\log ( {\vL(\vx,t_{ref})} \right|\geq c \ .
    \end{align}


  \end{frame}

  \begin{frame}{Introduction}{Event-based Cameras}
    \begin{figure}[c]

    \includegraphics[width=0.4\linewidth]{img/hawk_img.png}%\caption{frame-based}
    \includegraphics[width=0.4\linewidth]{img/hawk_events.png}%\caption{event-based}
    \end{figure}

  \end{frame}

  \begin{frame}{Introduction}{Optical Flow Estimation}
    % Optical flow refers to the instantaneous velocity of a pixel on an image of moving objects. It is caused by the relative movement between the camera and the objects.


    Omitted. Oh my bad English.

  \end{frame}

  \begin{frame}{Introduction}{Optical Flow Estimation}
    \begin{figure}
      \centering
      \includegraphics[width=0.3\textwidth]{img/fidget_dark.jpeg}
      \includegraphics[width=0.3\textwidth]{img/ball.jpeg}
      \includegraphics[width=0.3\textwidth]{img/waterfall_zoomed.jpeg}
    \end{figure}
  \end{frame}
  \begin{frame}
    \frametitle{Characteristics}

    \begin{itemize}
      \item High temporal resolution (milliseconds order)
      \item Asynchronous
      \item Sparse (spatial as well as temporal)
    \end{itemize}

    % High temporal resolution (milliseconds) and sparse data.
    % Event data can be either polar or non-polar.
  \end{frame}

  \begin{frame}
    \frametitle{Why this paper}

    \begin{itemize}
      \item Explanable
      \item Novel
      \item Effective
    \end{itemize}

  \end{frame}

  \section{Variational Approach}
  \begin{frame}
    \frametitle{Variational Approach} %\cite{...}

    \begin{equation}  \label{eq:opc}
      \vL(\vx,f) = \vL(\vx+\vu(\vx),t) \ ,
    \end{equation}

    Define $\vu=(u,v)$ to be an optical flow field, and $\vu(\vx) = (u_{\vx}, v_{\vx})^{\rmT}$ its value at pixel $\vx$. $\vu \in \R^{H \times W \times 2}$, and $\vL \in \R^{H\times W}$ is the latent image.

    \begin{equation}\label{eq:horn}
      \min_{\vu} \int_{\vX} \|\nabla\vu(\vx)\|^2 \, d\vx
      +\int_{\vX} (\vL(\vx,f)-\vL(\vx+\vu(\vx),t))^2 \,d\vx \ ,
    \end{equation}
  \end{frame}

  \section{Event-based Approach}
  \begin{frame}
    \frametitle{Event-Based Approach}

    % We therefore propose to jointly estimate flow ${\vu}$ and the latent image
    % ${\vL}$ by enforcing the brightness constancy by events and the blurred image formation model. In particular, our energy minimization model is formulated as:

    \begin{equation}
         \min_{\vL,\vu} \mu_1 \phi_\mathrm{eve}(\vL,\vu) +\mu_2
    \phi_\mathrm{blur}(\vL,\vu) +  \phi_\mathrm{flow}(\nabla\vu) +
    \phi_\mathrm{im}(\nabla\vL)
    \end{equation}


  \end{frame}



  \begin{frame}
    \frametitle{Brightness Constancy by Event Data $\phi_\mathrm{eve}$}
    Following the EDI model, this work present the neighbouring image as
    \begin{equation}\label{eq:Ltsigma}
      \begin{split}
      \vL(\vx,t)\
      & = \vL(\vx,f) \, \exp( c\,  \vE(\vx,t))\ , \\
      \end{split}
    \end{equation}

    Assume the motion between $\triangle t =t-f$ is small, we can adopt the first-order Taylor expansion on the right side of Eq.~\eqref{eq:opc} and obtain its approximation.

    \begin{equation}
      \begin{aligned}
      \vL(\vx & +\vu(\vx),f+\triangle t) \\
      & \approx \vL(\vx,f) + u_{\vx}\vL(\vx,f)^{(x)} +
      v_{\vx}\vL(\vx,f)^{(y)}+\triangle t \ \vL(\vx,f)^{(t)}\\
      &= u_{\vx}\vL(\vx,f)^{(x)} + v_{\vx}\vL(\vx,f)^{(y)} +\vL(\vx,t) \ . \\
      \end{aligned}
    \end{equation}
  \end{frame}

  \begin{frame}
    \frametitle{Brightness Constancy by Event Data $\phi_\mathrm{eve}$}
    \begin{equation}\nonumber
      \begin{split}
      \vL(\vx,f)
      \approx\  u_{\vx}\vL(\vx,f)^{(x)} + v_{\vx}\vL(\vx,f)^{(y)} +
      \vL(\vx,f)\exp\left(c\, \vE(\vx,t)\right).
      \end{split}
    \end{equation}
  \end{frame}

  \begin{frame}
    \frametitle{}

    \begin{eqnarray}\label{eq:energyeve}
      \boxed{
      \begin{aligned}
      \phi_\mathrm{eve}(\vL,\vu)
      =\sum_{\vx \in \vX}\| & \vL(\vx,f)(\exp( c\, \vE(\vx,t))-1)\\
      &+[u_{\vx},v_{\vx}]^{\rmT}\nabla {\vL}(\vx,f) \|_1 \ .
      \end{aligned}
      }
      \end{eqnarray}

  \end{frame}

  \begin{frame}
    \frametitle{Blur Image Formation Constraint $\phi_\mathrm{blur}$}

    A general model of blur image formation is given as
    \begin{eqnarray} \label{eq:bm}
      \vB =  \vk \otimes \vL(f) \ ,
    \end{eqnarray}


    %{\small
\begin{equation} \label{eq:pixel-bm0}
  \begin{aligned}
  \vB(\vx) & = \vk(\vx)\otimes \vL(\vx)\ .\\
  \end{aligned}
\end{equation}
  \end{frame}

  \begin{frame}
    \frametitle{}

  We omit $f$ in the following sections. The convolution of the two matrices is
  defined as,
  \vspace*{-1mm}
  \begin{equation} \label{eq:pixel-bm}
  \begin{aligned}
  \vB(\vx) &= \sum_{\vy \in \vX} \vk(\vy) \vL(\vx-\vy)\\
  &= \sum_{\vy \in \vX} \vk_{\vu'(\vx)}(\vy) \vL(\vx-\vy) \ ,
  \end{aligned}
  \end{equation}

  where

  \begin{equation}
    \begin{aligned} \label{eq:kimblurKernel}
     k_{\vu'(\vx)}&(\vy)%\\
    =&\left\{
    \begin{gathered}
    \begin{aligned}
    &\frac{1}{|\vu'(\vx)|},
    &\;{\rm if}\;\vy=\alpha\vu'(\vx),\ |\alpha|\leq\frac{1}{2}\\
    & {\bf 0}, & {\rm otherwise}\ ,
    \end{aligned}
    \end{gathered}\right.
    \end{aligned}
    \end{equation}

  \end{frame}

  \begin{frame}
    \frametitle{}

    \begin{equation}\label{eq:phicolor}
      \boxed{
      \phi_\mathrm{blur}(\vL,\vu) =\sum_{\vx,\vy \in \vX} \|\vk_{\vu'(\vx)}(\vy)
      \vL(\vx-\vy)-\vB(\vx)\|^2 \ ,
      }
      \end{equation}

  \end{frame}

  \begin{frame}
    \frametitle{}

    \begin{equation}
      \boxed{
      \begin{aligned}
      \phi_\mathrm{flow}(\nabla \vu)
      =\|w\nabla \vu\|_{1,2}
      = \sum_{\vx \in \vX}\|w(\vx)\nabla \vu(\vx)\|\ ,
      \end{aligned}
      }
      \end{equation}


      \begin{equation}
        \boxed{
        \phi_\mathrm{im}(\nabla \vL) = \sum_{\vx \in \vX}\|\nabla\vL(\vx)\|_1 \ .
        }
        \end{equation}

  \end{frame}

  \begin{frame}
    \frametitle{}
    \centering
    Thank you for listening!\footnote{Presentation prepared using \LaTeX{} Beamer}

  \end{frame}
\end{document}

